How to smooth or blur a procedural texture?
To blur such a procedural texture, you can't simply add a 'blur' node or similar at the point you indicate. Instead, you need to manipulate the input Vector to distort the texture and then rely on the Render Samples to produce the blurring.

One easy way of achieving this is to add the nodes shown.



Use the relevant output of the Texture Coordinate node (Generated, Object, UV, etc) depending on how you want the texture (and blurring) to be applied. The 'Subtract' node is set to its default (mid-gray) with factor set to 1.000 - this is to adjust the Noise texture to ensure the blurring is even. The Noise texture is set to a very high scale and the Factor of the Add node can be adjusted to change the strength of the blur (0 = no blur, 1.0 is maximum blur).

As mentioned, you should set your Render Samples (on the Render properties) to a higher value so as to allow the rendering enough samples to produce a clean 'blur'.



Examples showing no blurring and blurring :